Ah.
- disable posting in mybb
- go to cpanel on your old host. Backups --> Home Directory/MySQL Database/Forwarders. Download each of them
- go to new cpanel. Click Backups --> beside where you downloaded backups, click restore and restore Home Directory/MySQL Database/Forwarders

Go to MySQL on the main page. Create a user. There is a config file somewhere in mybb that has the user/pass for the database. Use that file to make a new user with the same pass. Then assign it to the database you restored.

Oh and change your name servers. Wait a few hours then enable posting.